Why should she tell the right away? Its not like she needs a reference from them, and they haven't been in any rush to communicate or worry about OPs needs. All of this is their fault, so any rushing to find childcare is their consequence to bear.
She should tell them right away because it is OP who is at fault. The family hired her and she accepted months ago. She is still a month from starting and just waiting on a final work agreement, which means all the negotiations are done. Why does she need this a month before a start date to have the final document? Also, why is it unusual for a family to not be in frequent contact with a nanny who isn't starting for months? The family isn't being flaky. OP is. The family hired *early* to take care of their child care needs. If OP needed to have the final document a month before starting, she should have warned them that if they didn't meet her date, she would need to find other work. She didn't do that. So, she is already screwing them over by changing her mind late (for them). So she owes it to them to let them know now.
Besides, OP isn't leaving because the family is flaky. She is leaving because she needs a job that pays better because her financial situation changed. She is just trying to make it their fault by saying they aren't communicating with her.